Chapter 8.4, Problem 46E

To determine

To find:

The determinant of the matrixand the cofactors usingby the given row and column

Answer to Problem 46E

Cofactors $C_{31}=224$

Explanation of Solution

Given:

The matrix

  $A=\left[\begin{array}{cccc}10& 8& 3& -7\\ 4& 0& 5& -6\\ 0& 3& 2& 7\\ 1& 0& -3& 2\end{array}\right]$

And the row $=3$

column $=1$

Concept used:

Minors and Cofactors of a square matrix:-

If A is a square matrix then the minor $M_{ij}$ of the entry $a_{ij}$ is the determinant of the matrix obtained by deleting the $i\text{th}$ row and $j\text{th}$ column of $A$

The cofactor $C_{ij}$ of the entry $a_{ij}$ is

  $C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Calculation:

The given matrix

  $A=\left[\begin{array}{cccc}10& 8& 3& -7\\ 4& 0& 5& -6\\ 0& 3& 2& 7\\ 1& 0& -3& 2\end{array}\right]$

row $=3$

column $=1$

If A is a square matrix then the minor $M_{ij}$ of the entry $a_{ij}$ is the determinant of the matrix obtained by deleting the $i\text{th}$ row and $j\text{th}$ column of $A$

  $\begin{array}{l}{M}_{31}=\left|\begin{array}{ccc}8& 3& -7\\ 0& 5& -6\\ 0& -3& 2\end{array}\right|\\ M_{31}=8\left(10+18\right)+3\left(0-0\right)+7\left(0-0\right)\\ M_{31}=224\end{array}$

The cofactor $C_{ij}$ of the entry $a_{ij}$ is

  $\begin{array}{l}{C}_{ij}={\left(-1\right)}^{i+j}{M}_{ij}\\ C_{31}={\left(-1\right)}^{3+1}{M}_{31}\\ C_{31}=M_{31}\\ C_{31}=224\end{array}$

Cofactors $C_{31}=224$